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August 2018, 12(4): 955-970. doi: 10.3934/ipi.2018040

## Inverse source problems without (pseudo) convexity assumptions

 1 Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA 2 School of Mathematical Sciences, and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Shuai Lu

Received  June 2017 Revised  January 2018 Published  June 2018

Fund Project: This research is supported in part by the Emylou Keith and Betty Dutcher Distinguished Professorship and the NSF grant DMS 15-14886. Shuai Lu is supported by NSFC No.11522108, 91630309, Shanghai Municipal Education Commission No.16SG01 and Special Funds for Major State Basic Research Projects of China (2015CB856003). The authors thank an anonymous referee for his careful reading of the manuscript and valuable remarks which greatly helped to improve the article

We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.

Citation: Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
##### References:
 [1] H. Ammari, G. Bao and J. Fleming, Inverse source problem for Maxwell's equation in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927. [2] C. A. Balanis, Antenna Theory-Analysis and Design, Wiley, Hoboken, NJ, 2005. [3] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465. doi: 10.1016/j.jde.2010.08.013. [4] G. Bao, J. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math., 548 (2011), 45-60. doi: 10.1090/conm/548/10835. [5] G. Bao, S. Lu, W. Rundell and B. Xu, A recursive algorithm for multi-frequency acoustic inverse source problems, SIAM J. Numer. Anal., 53 (2015), 1608-1628. doi: 10.1137/140993648. [6] J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804. doi: 10.1016/j.jde.2015.11.030. [7] M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems, 25 (2009), 115005 (20pp). doi: 10.1088/0266-5611/25/11/115005. [8] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and Stability in the Cauchy Problem for Maxwell' and elasticity systems, Nonlinear Partial Differential Equations and Their Applications (D Cioranescu and J.-L. Lions, eds.), North-Holland, Elsevier Science, 31 (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9. [9] V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., 426 (2007), 255-267. doi: 10.1090/conm/426/08192. [10] V. Isakov, On increasing stability in the Cauchy Problem for general elliptic equations, New Prospects in Direct, Inverse, and Control Problems for Evolution Equations Ch. 10. Springer INdAM Series. (A. Favini et al., ed.), North-Holland, Elsevier Science, Springer-Verlag, 2014. [11] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-51658-5. [12] V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. Anal., 18 (2011), 1-29. doi: 10.4310/MAA.2011.v18.n1.a1. [13] V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Problems and Imaging, 8 (2014), 1139-1150. doi: 10.3934/ipi.2014.8.1139. [14] V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594. doi: 10.1137/15M1019052. [15] V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1-18. doi: 10.1137/17M1112704. [16] F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin-Heidelberg, 1972. [18] T. Kato, Perturbation Theory for Linear Operators, Band 132 Springer-Verlag New York, Inc., New York, 1966. [19] D. Tataru, Unique continuation for solutions to PDE's: Between Hörmander's Theorem and Holmgren's Theorem, Comm. Part. Diff. Equat., 20 (1995), 855-884. doi: 10.1080/03605309508821117. [20] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1922.

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##### References:
 [1] H. Ammari, G. Bao and J. Fleming, Inverse source problem for Maxwell's equation in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927. [2] C. A. Balanis, Antenna Theory-Analysis and Design, Wiley, Hoboken, NJ, 2005. [3] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465. doi: 10.1016/j.jde.2010.08.013. [4] G. Bao, J. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math., 548 (2011), 45-60. doi: 10.1090/conm/548/10835. [5] G. Bao, S. Lu, W. Rundell and B. Xu, A recursive algorithm for multi-frequency acoustic inverse source problems, SIAM J. Numer. Anal., 53 (2015), 1608-1628. doi: 10.1137/140993648. [6] J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804. doi: 10.1016/j.jde.2015.11.030. [7] M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems, 25 (2009), 115005 (20pp). doi: 10.1088/0266-5611/25/11/115005. [8] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and Stability in the Cauchy Problem for Maxwell' and elasticity systems, Nonlinear Partial Differential Equations and Their Applications (D Cioranescu and J.-L. Lions, eds.), North-Holland, Elsevier Science, 31 (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9. [9] V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., 426 (2007), 255-267. doi: 10.1090/conm/426/08192. [10] V. Isakov, On increasing stability in the Cauchy Problem for general elliptic equations, New Prospects in Direct, Inverse, and Control Problems for Evolution Equations Ch. 10. Springer INdAM Series. (A. Favini et al., ed.), North-Holland, Elsevier Science, Springer-Verlag, 2014. [11] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-51658-5. [12] V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. Anal., 18 (2011), 1-29. doi: 10.4310/MAA.2011.v18.n1.a1. [13] V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Problems and Imaging, 8 (2014), 1139-1150. doi: 10.3934/ipi.2014.8.1139. [14] V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594. doi: 10.1137/15M1019052. [15] V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1-18. doi: 10.1137/17M1112704. [16] F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin-Heidelberg, 1972. [18] T. Kato, Perturbation Theory for Linear Operators, Band 132 Springer-Verlag New York, Inc., New York, 1966. [19] D. Tataru, Unique continuation for solutions to PDE's: Between Hörmander's Theorem and Holmgren's Theorem, Comm. Part. Diff. Equat., 20 (1995), 855-884. doi: 10.1080/03605309508821117. [20] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1922.
Domain of the source problem. The source function is compactly supported in $\Omega\setminus\Omega_0$
Annular domain of the interior inverse source problem. Solid red line is the observation $\Gamma$ and the domain $\Omega$ is $B_{2}(0.3, 0.3)\setminus \bar B_{0.5}(0, 0)$
Annular domain with a piecewise constant source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wavenumber $k$ with the minimal relative error $0.1856$ at $k = 200$
Annular domain with a mixed-point source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.1795$ at $k = 200$
Annular domain with different observation sites for exact measurement data. $\Gamma = (0.3+2\cos\theta, 0.3+2\sin\theta)$, $\theta\in (0, \pi)$; $\Gamma_1 = (0.3+5\cos\theta, 0.3+5\sin\theta)$, $\theta\in(0, \frac{2}{5}\pi)$ and $\Gamma_2 = (0.3+10\cos\theta, 0.3+10\sin\theta)$, $\theta\in (0, \frac{1}{5}\pi)$. Left: the relative error slope versus the increasing wave number for the piecewise constant source; Right: the relative error slope versus the increasing wave number for the mixed-point source. The values at the tail of each line are the minimal relative errors with $k = 200$ or $k = 400$
Rectangular domain of the interior inverse source problem. Solid red line is the observation site $\Gamma = \{0.45\}\times [-0.45, 0.45]$ and the blue shadowed domain $(0.5, 2.5)\times (-1, 1)$ is the source domain which is a subset of $\Omega$
Rectangular domain with piecewise constant sources for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.3149$ at $k = 200$
Error slopes for fixed observation site $\Gamma$ but different recovery domains. The values at the tail of each line are the minimal relative errors with $k = 200$ or $k = 400$
Annular domain of the exterior inverse source problem. Solid red line is the observation site $\Gamma$ and shadowed domain is $\Omega$
Annular domain with a piecewise constant source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.3923$ at $k = 200$
Rectangular domain of the exterior inverse source problem. Solid red line is the observation site $\Gamma$ and shadowed domain is $\Omega$
Rectangular domain with piecewise constant sources for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.1747$ at $k = 200$
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